Stable pairs Virasoro constraints for simply connected smooth projective 3-folds

Let $X$ be a simply-connected smooth projective 3-fold. Let $\mathbb D_{\textup{PT}}^{X}$ be the algebra generated by stable-pairs descendents $\textup{ch}_i(\gamma)$, and let $\mathcal L_k=R_k+T_k+S_k$ be the Virasoro operator defined on this algebra for $k\geq -1$. For $D\in\mathbb D_{\textup{PT}}^{X}$, write $\langle D\rangle_{n,\beta}^{X,\textup{PT}}$ for integration over the stable-pairs virtual class. Stable pairs Virasoro conjecture. For all $k\geq -1$, $\beta\in H_2(X;\mathbb Z)$, $n\in\mathbb Z$, and $D\in\mathbb D_{\textup{PT}}^{X}$,

$$\left\langle\mathcal L_k(D)\right\rangle_{n,\beta}^{X,\textup{PT}}=0.$$

These constraints are the stable-pairs analogue of the Virasoro constraints in Gromov–Witten theory. They are known for toric 3-folds when $D$ is stationary, while the stated result for arbitrary simply-connected smooth projective 3-folds remains open.